A finite-difference scheme for initial boundary value problem of the Gamma equation in the pricing of financial derivatives

Abstract

In the article, we consider the initial boundary value problem for the Gamma equation, which can be derived by transforming the nonlinear Black-Scholes equation for option price into a quasilinear parabolic equation for the second derivative of the option price. We develop unconditionally monotone finite-difference schemes of second-order of local approximation on uniform grids for the initial boundary value problem for the Gamma equation. Two-side estimates of the solution of the scheme are established. By means of regularization principle, the previous results are generalized for construction of unconditionally monotone finite-difference scheme (the maximum principle is satisfied without constraints on relations between the coefficients and grid parameters) of the second-order of approximation on uniform grids for this equation. With the help of difference maximum principle, the two-side estimates for difference solution are obtained at the arbitrary non-sign-constant input data of the problem. A priori estimate in the maximum norm C is proved. It is interesting to note that the proven two-side estimates for difference solution are fully consistent with the differential problem, and the maximal and minimal values of the difference solution do not depend on the diffusion and convection coefficients. Computational experiments, confirming the theoretical conclusions, are given.

A Fuzzy Random Boundary Value Problem

“…Finally, we prove the existence of a unique random solution (EURS) for system (2). In fact, our work is in a fuzzy state of [7] (see also [18][19][20][21]).…”

Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

“…We can construct similar schemes by using the identities (28) and ku = 0.5 (ku) + ku − k u . Suppose that b (x) = r (x) /k (x) and L 1 v = v + bv we rewrite the equation (42) in the form…”

“…Not only in mathematical physics, but also in economics, there is a need to solve partial differential equations containing lower derivatives. For example, in financial mathematics, it is of interest to study the Gamma equation obtained by a transformation of the nonlinear Black-Scholes equation to a quasilinear parabolic equation [26], [27], [30], [42]. The approximate solution of the Gamma equation is the main goal of this study.…”